The claim we wish to prove is that if p≤q, then θ(p)≤θ(q) -- increasing the probability that an edge is open will not decrease the probability of finding an infinite cluster. This is intuitively obvious: it feels like we only open more edges at q than we do at p, so naturally clusters can only grow.

Unfortunately the proof is a bit more involved. The problem is that the probability space Ωp for percolation on G with edges open with probability p is a different probability space than Ωq, the one for percolation on G with edges open with probability q. It is not clear how to transfer events from one probability space to another.

In other words: We assign a random number 0≤xe≤1 to each edge e. We transform this assignment into percolation with probability p by considering an edge open in Ωp if the random value assigned to in Ω is between 0 and p.

Obviously this describes percolation on Ωp. Equally obvious, this is a measure-preserving transformation.
By the coupling argument presented above, the intuitively obvious -- that the percolation probability increases with the edge probability -- is proved.